MEMO 2025 I1

Let $\mathbb{R}^+$ be the set of positive real numbers. Let $f: \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that for all $x, y \in \mathbb{R}^+$ it holds that

$$yf^{2025}(x) \geq xf(y).$$

Show that there exists a positive integer $n_0$ such that for all positive integers $n \geq n_0$ and for all $x \in \mathbb{R}^+$ it holds that

$$f^n(x) \geq x.$$

Remark. Here $f^n$ denotes the function $f$ applied $n$ times, this means $f^n(x) = \underbrace{f(f(\ldots f(x) \ldots))}_{n \text{ times}}.$